PHYSICAL REVIEW B 90, 075138 (2014)

Competing orders, competing anisotropies, and multicriticality: The case of Co-doped YbRh2Si2

Eric C. Andrade,1,2 Manuel Brando,3 Christoph Geibel,3 and Matthias Vojta1

1Institut fur Theoretische Physik, Technische Universitat Dresden, 01062 Dresden, Germany2Instituto de Fısica Teorica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz, 271 - Bl. II, 01140-070, Sao Paulo, SP, Brazil

3Max-Planck-Institut fur Chemische Physik fester Stoffe, Nothnitzer Straße 40, 01187 Dresden, Germany(Received 20 December 2013; revised manuscript received 11 August 2014; published 22 August 2014)

Motivated by the unusual evolution of magnetic phases in stoichiometric and Co-doped YbRh2Si2, we studyHeisenberg models with competing ferromagnetic and antiferromagnetic ordering combined with competinganisotropies in exchange interactions and g factors. Utilizing large-scale classical Monte Carlo simulations, weanalyze the ingredients required to obtain the characteristic crossing point of uniform susceptibilities observedexperimentally near the ferromagnetic ordering of Yb(Rh0.73Co0.27)2Si2. The models possess multicritical points,which we speculate to be relevant for the behavior of clean as well as doped YbRh2Si2. We also make contactwith experimental data on YbNi4P2, where a similar susceptibility crossing has been observed.

DOI: 10.1103/PhysRevB.90.075138 PACS number(s): 64.60.Kw, 71.27.+a, 75.10.Hk, 75.30.Gw

I. INTRODUCTION

Magnetic quantum phase transitions (QPTs) in solids are anactive area of research, with many unsolved problems presentparticularly for metallic systems [1,2]. While the simplesttheoretical models assume a single type of ordering instabilityand SU(2) spin rotation symmetry, real materials often possessadditional ingredients which complicate the interpretation ofexperimental data: (i) a competition of multiple ordered states,e.g., with different ordering wave vectors and/or differentspin structures that may result from geometric frustration ormultiple nesting conditions, and (ii) magnetic anisotropies thatcan modify ordered states and induce additional crossoverenergy scales.

The tetragonal heavy-fermion metal YbRh2Si2 is a promi-nent example for anisotropic and competing magnetic orders.Its uniform magnetic susceptibility is an order of magnitudelarger for fields Bab applied in the basal plane as compared tofields Bc along the c axis. Stoichiometric YbRh2Si2 displaysan ordered phase below TN = 70 mK, believed to be anantiferromagnet with moments oriented in the basal plane.This order is destroyed at a field-driven QPT at Bcrit

ab = 60 mT;the corresponding c -axis critical field is Bcrit

c = 0.66 T [3,4].Upon applying hydrostatic pressure or doping with Co,

both TN and Bcrit increase which is primarily caused by areduction of Kondo screening. Moreover, a second phasetransition at TL < TN occurs at small fields, Fig. 1(a), withthe precise nature of the phase for T < TL being unknown[5,6]. Interestingly, an almost divergent uniform susceptibilitywas reported for stoichiometric YbRh2Si2, hinting at thepresence of ferromagnetic correlations [7]. However, in-planeferromagnetic order could be detected neither under pressurenor with doping.

Given this state of affairs, it came as a surprise thatYb(Rh0.73Co0.27)2Si2 was recently found [8] to develop fer-romagnetic order below TC = 1.3 K with moments orientedalong the c axis, which was identified as the hard mag-netic axis from all previous measurements. This “switch”of response anisotropies in Yb(Rh0.73Co0.27)2Si2 is reflectedin the susceptibilities, where χab(T ) and χc(T ) displaya characteristic crossing point at TX slightly above TC

[Fig. 1(b)]. Furthermore, the evolution of TL with doping in

Yb(Rh1−xCox)2Si2 [Fig. 1(a)] suggests that the transition at TL

involves a ferromagnetic ordering component for x < 0.27.Parenthetically, we note that antiferromagnetism takes overagain for x � 0.6, with YbCo2Si2 displaying a Neel transitionat TN = 1.65 K and a large in-plane ordered moment of1.4 μB [6,9].

Taken together, the data show that the magnetism ofYb(Rh1−xCox)2Si2 is determined by competing antiferromag-netic (AFM) and ferromagnetic (FM) orders, which, moreover,are characterized by competing anisotropies.

In this paper we propose that Heisenberg models of localmoments can account for many of the unusual magnetic prop-erties of Yb(Rh1−xCox)2Si2 by invoking multiple magneticinteractions and a competition between anisotropic exchangeand anisotropic g factors. We primarily focus on the behaviorof Yb(Rh0.73Co0.27)2Si2 in the vicinity of its finite-temperaturephase transition which can be modeled classically. For thesimplest models, we investigate thermodynamic properties us-ing large-scale Monte Carlo (MC) simulations and provide anexplicit comparison to experimental susceptibility data. Basedon our results, we propose that multicritical points, naturallypresent in our modeling, are of relevance for understandingthe unusual quantum criticality in stoichiometric YbRh2Si2.Finally, we also discuss the behavior of the heavy-fermionferromagnet YbNi4P2, which displays a similar “switch” ofmagnetic response anisotropy [10,11].

The body of the paper is organized as follows: In Sec. IIwe describe the family of anisotropic Heisenberg models tobe studied, together with methodological aspects of the MCsimulations. Section III is devoted to a detailed modeling ofYb(Rh0.73Co0.27)2Si2. It starts with a discussion of competingmagnetic anisotropies and then introduces competing FM andAFM ordering tendencies, in order to match experimentalobservations. In Sec. IV we then discuss the presence ofmulticritical points in the phase diagrams of our models andspeculate on the role of quantum bicriticality in the phasediagram of YbRh2Si2. Finally, in Sec. V we briefly discuss thecase of YbNi4P2. A short summary concludes the paper. In theAppendix we illustrate the effects of an additional single-ionanisotropy, not present in the spin-1/2 models discussed in thebulk of the paper.

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ANDRADE, BRANDO, GEIBEL, AND VOJTA PHYSICAL REVIEW B 90, 075138 (2014)

FIG. 1. (Color online) (a) Temperature-composition phase dia-gram of YbRh2Si2 showing the two magnetic transitions at TN andTL. Pressure (lower axis) has an effect similar to Co doping (upperaxis). (We note that ongoing studies suggest an even more complexphase diagram with canted AFM order below TL for x < 0.2.) (b)Susceptibility data from Yb(Rh0.73Co0.27)2Si2 showing the switch ofthe anisotropy of the magnetic response as function of temperature.Figure reproduced from Ref. [8].

II. MODEL AND SIMULATIONS

A. Model and magnetic anisotropies

The Yb3+ ions in YbRh2Si2 are in a 4f 13 configurationin a tetragonal crystal field. The ionic ground state is a�7 Kramers doublet [12], which can be represented by aneffective (pseudo)spin-1/2 moment, which further couples toconduction electrons via a Kondo exchange coupling. Due tostrong spin-orbital coupling, SU(2) spin symmetry is brokenand magnetic anisotropies appear: (i) The (pseudo)spinscouple to an external Zeeman field in an anisotropic fashion,leading to an anisotropic g tensor. (ii) Magnetic interactionsbetween the moments will be anisotropic in spin space,which applies to both direct exchange and Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions. A single-ion anisotropyis forbidden for elementary spin-1/2; for completeness wediscuss its effects in models of spins S � 1 in the Appendix.

In this paper, we consider spin-only models of Yb momentsin YbRh2Si2. These models will be of Heisenberg type; toaccount for competing phases, we will include longer-range

exchange interactions. For simplicity, we place the momentson a cubic lattice and consider models of the form

H = −∑

〈i,j〉αJ α

1 Sαi Sα

j −∑

〈〈i,j〉〉αJ α

2 Sαi Sα

j −∑

iα

gαhαSαi , (1)

where J1 and J2 denote couplings to first and second neighbors.Sα

i is the α = x,y,z component of the moment at site i, hα

represents the external magnetic field, and gα are the (diagonal)components of the g tensor. Because the Yb moments are in atetragonal crystal field, we have gx = gy ≡ gab and gz ≡ gc.Analogously, the exchange interaction between the Yb3+ ionscan be parameterized as J x

1 = Jy

1 = J1(1 + �1) and J z1 ≡

J1. For simplicity, we consider isotropic second-neighborcouplings, with J α

2 = J2. Being interested in the behavior nearthe finite-temperature phase transition, we approximate thequantum spins as classical vectors with | �Si | = 1.

B. Monte Carlo simulations

To access the finite-temperature behavior of Eq. (1) weperform equilibrium MC simulations on cubic lattices ofsize N = L × L × L, with L � 32 and periodic boundaryconditions. We employ the single-site Metropolis algorithmcombined with microcanonical steps to improve the samplingat lower temperatures. Typically, we run 106 MC steps(MCS) as initial thermalization followed by 106 MCS toobtain thermal averages, which are calculated by dividing themeasurement steps into 10 bins. Here one MCS corresponds toone attempted spin flip per site. We consider the same numberof Metropolis and microcanonical steps. In all our results weset kB = 1 and a = 1, where kB is the Boltzmann constant anda is the lattice spacing and use J1 as our energy scale.

From the MC data, we calculate thermodynamic ob-servables as the specific heat C and the uniform magneticsusceptibilities, both along the c axis,

χc(T ) = g2c

N

T

(⟨m2

c

⟩ − 〈mc〉2), (2)

and in the ab plane,

χab(T ) = 1

2g2

ab

N

T

⟨m2

ab

⟩, (3)

where 〈· · · 〉 denotes the MC average, mc = Mz, and mab =(M2

x + M2y )1/2 with Mα = N−1 ∑

i Sαi . Due to its definition,

mab incorporates the fact that the x and y components of �Sare equivalent. In χab no subtraction of expectation values isnecessary, as we will only consider ferromagnetic states withorder along the c axis. In the linear-response limit, g factors donot enter the MC simulations directly but appear as prefactorsin Eqs. (2) and (3) only.

Further, we analyze phase transitions via the Bindercumulant [13] and we characterize ordered states by the static

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COMPETING ORDERS, COMPETING ANISOTROPIES, AND . . . PHYSICAL REVIEW B 90, 075138 (2014)

100 101

T/J1

10−2

10−1

100

101

102

χ(T

)

(a)

χc

χab

100 101

T/J1

(b)

100 101

T/J1

(c)

FIG. 2. (Color online) Anisotropic susceptibilities χab (green) and χc (blue) on a log-log scale for FM (J x,y > 0) spin exchange in theab plane, J2 = 0, gab = gc, and L = 16: (a) � = −0.05; (b) � = −0.10; and (c) � = −0.50. Finite-size effects are negligible except fortemperatures within 0.2% of TC.

spin structure factors:

Sc(�q) = 1

N

∑

i,j

e−i �q·�rij⟨Sz

i Szj

⟩, (4)

Sab(�q) = 1

N

∑

i,j

e−i �q·�rij⟨Sx

i Sxj + S

y

i Sy

j

⟩. (5)

In a long-range ordered phase Sc(ab)(�q)/N → m2c(ab)δ�q, �Q,

where δ is the Kronecker delta and �Q is the orderingwave vector. Note that mc(ab) → 1 as T → 0, because theclassical ground state is fluctuationless. The behavior ofSc(ab)(�q) near �Q allows one to extract the correlation lengthξc(ab)(T ) characterizing the magnetic order. In our finite-sizesimulations, the ordering temperature for the second-orderphase transition TC is efficiently extracted from the crossingpoints of ξ (T )/L data for different L, according to the scalinglaw ξ (T )/L = f [L1/ν(T − TC)], where f (x) is a scalingfunction and ν is the correlation length exponent.

III. MODELING OF Yb(Rh0.73Co0.27)2Si2

In this section, we describe a stepwise modeling of mag-netism in Yb(Rh0.73Co0.27)2Si2 using spin models of the form(1), with the goal of reproducing the temperature dependenceof χ (T ) in Fig. 1(b). Conduction electrons are not part of thismodeling, i.e., we assume that the magnetic properties canbe described in terms of (effective) local-moment models.We note that this assumption is not in contradiction withpossible Kondo screening of the Yb 4f moments: First, inKondo-lattice systems the local-moment contributions to themagnetic response are typically much larger than those ofthe conduction electrons. Second, universality dictates that,even in cases where magnetism is fully itinerant, local-momentmodels successfully describe many long-wavelength magneticproperties [14].

A. Competing anisotropies and bicriticality

A central observation in the phase diagram [6,8] ofYb(Rh1−xCox)2Si2 [Fig. 1(a)] is that the two thermal phasetransitions at TN and TL, existing for small x, merge close tox = 0.27. Combining the facts that TN marks a transition intoan in-plane antiferromagnet and the state below TC at x = 0.27is a c-axis ferromagnet suggests the competition of these twoinstabilities, with a proximate bicritical (or, more generally,multicritical) point.

We start by focusing on the competition between in-planeand c-axis order, which can be captured, in a spin-1/2model as in Eq. (1), by anisotropic exchange couplings,with |J x,y

1 | = |J z1 |(1 + �1). Degeneracy of the two orders

is trivially reached at the Heisenberg point �1 = 0, wherethe transition corresponds to a bicritical point. A transitionto c-axis order as in Yb(Rh0.73Co0.27)2Si2 requires �1 < 0.Sample susceptibilities results assuming FM (J x,y > 0) spinexchange in the ab plane are shown in Fig. 2, where we see thatwhile χc(T ) is essentially independent of the anisotropy �1,χab(T ) is considerably enhanced as we approach the bicriticalpoint, due to the increase in the in-plane fluctuations. Thebehavior of the magnetic susceptibilities in the vicinity of abicritical point is further explored in Appendix A.

The fact that, experimentally, χc � χab for a large rangeof temperatures above the transition (essentially from TC up to100 K) implies gc < gab, i.e., the g-factor anisotropy is oppo-site to the exchange anisotropy. Indeed, for undoped YbRh2Si2a strong g-factor anisotropy of gab/gc ≈ 20 has been deducedfrom electron-spin-resonance (ESR) experiments [12]. ForYb(Rh0.73Co0.27)2Si2 we have [8] gab/gc = 6.4, and we notethat the end member of the doping series, YbCo2Si2, has asmaller anisotropy, gab/gc ≈ 2.5, indicating a considerablevariation of the g-factor anisotropy ratio throughout the series,even though no change in the symmetry of the ground-statedoublet is observed [9,15].

In a modeling of the susceptibility data ofYb(Rh0.73Co0.27)2Si2 under the assumption of exclusivelynearest-neighbor ferromagnetic coupling J α

1 > 0 (thisassumption will be relaxed in Sec. III B below) we are leftwith the following free parameters: the overall energy scaleJ1 which determines TC, the exchange anisotropy �, and theg-factor anisotropy gab/gc. (The absolute value of χ , or g, issimply adjusted to the data by matching the low-T value ofχab.) To determine � and gab/gc different strategies appearpossible; we found the following useful: First, � is chosento match the temperature dependence of the nondivergentχab(T ) near TC (where χc diverges). In general, small �

should be chosen to ensure proximity to bicriticality. Second,with � fixed, the crossing temperature TX of χab and χc canbe used to determine the g-factor anisotropy by demandingthat experiment and theory yield the same TX/TC ratio;experimentally this number is TX/TC = 1.016. After thisprocedure is implemented, we add a constant susceptibilityχv to mimic the Van Vleck contribution arising from higher

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ANDRADE, BRANDO, GEIBEL, AND VOJTA PHYSICAL REVIEW B 90, 075138 (2014)

10−1

100

101

102χ

(T)

(a)χab, theoryχab, Ref. 8χc, theoryχc, Ref. 8

10−1

100

101

102

χ(T

)

(b)

10−2 10−1 100 101 102

T/TC

10−1

100

101

102

χ(T

)

(c)

FIG. 3. (Color online) Comparison between the numerical andexperimental susceptibilities χ (T ) as a function of T/TC on a log-logscale. The continuous curves show the simulation results, χab in greenand χc in blue, obtained for � = −0.05 and L = 32 in Eq. (1), withdifferent exchange parameters in panels (a)–(c). The dashed lines withsymbols represent the corresponding experimental susceptibilitiesreported in Ref. [8] in units of 10−6 m3/mol, with TC = 1.30 K.(a) FM (J x,y > 0) spin exchange in the ab plane, with gab/gc =2.5 and J1 = 0.87 K. (b) Same as (a), but for AFM (J x,y < 0) spinexchange in the ab plane and gab/gc = 16.4. (c) Same as (b), withadditional isotropic next-nearest-neighbor FM coupling J2, with J2 =J1, J1 = 0.25 K, and gab/gc = 7.3. In all simulation results, we haveadded a constant Van Vleck background χv = 0.20 × 10−6 m3/mol,as is seen in Yb(Rh0.73Co0.27)2Si2 [8].

crystal-electric-field levels which are not part of our model.The value chosen is χv = 0.20 × 10−6 m3/mol, which isclose to the value of 0.172 × 10−6 m3/mol deduced forYb(Rh0.73Co0.27)2Si2 [8].

For exclusively nearest-neighbor ferromagnetic coupling,the first step cannot be satisfactorily followed: The temperaturevariation of χab is too large compared to experiment. A sampleset of data is presented in Fig. 3(a). Here � = −0.05 hasbeen chosen as a compromise; fixing TX/TC then yieldsgab/gc = 2.5. The poor agreement with experiment will becured upon considering dominant in-plane antiferromagneticcorrelations below.

B. Competing ferromagnetism and antiferromagnetism

While the significant increase of χab towards low tem-peratures in both Co-doped and undoped YbRh2Si2 re-flects the presence of ferromagnetic exchange, the tendency

towards antiferromagnetism cannot be ignored. Since un-doped YbRh2Si2 is believed (and YbCo2Si2 was proved)to display low-temperature antiferromagnetic order, it isvery likely that strong in-plane antiferromagnetism exist inYb(Rh0.73Co0.27)2Si2 as well.

First we repeat the analysis described in Sec. III Awith purely AFM in-plane exchange, i.e., we consider themodel (1) with J

x,y

1 = −J1(1 + �1) < 0 and J z1 = J1 > 0.

Corresponding results are presented in Fig. 3(b). Since thecoupling of the spins’ z components remains FM, χc does notchange as compared to Fig. 3(a). In contrast, χab is not onlystrongly suppressed but also shows much less temperaturevariation, as expected for an antiferromagnet. For T > TC

the susceptibility follows roughly χab ∝ 1/(T − ab) with aCurie-Weiss temperature ab ∼ J x,y < 0, which results in aflattening towards low temperature. To restore the χ crossing atthe experimental value of TX/TC, we now need to assume a g-factor ratio of gab/gc = 16.4. The behavior of χab for T < TC

is now much closer to that observed in Yb(Rh0.73Co0.27)2Si2.However, its very slow decay upon heating above TC stillindicates a rather poor match with the experiment.

The natural conclusion is that both FM and AFM tendenciesare present for the in-plane order, a qualitative conclusionreached earlier on the basis of experimental data [7]. Thereare various ways to implement the competition of FM andAFM order in a microscopic spin model. Here we choosea simple and frustration-free way. We employ an isotropicferromagnetic second-neighbor coupling J2. For J

xy

1 = 0 andJ2 �= 0, we have two decoupled sublattices which indepen-dently order ferromagnetically. As first discussed in Ref. [16],thermal fluctuations select collinear states such that the wavevectors �Q = (0,0,0) (sublattices parallel) and �Q = (π,π,π )(sublattices antiparallel) are degenerated. A finite J

xy

1 ≷ 0can then be used to tip the balance between dominantferromagnetic and antiferromagnetic in-plane correlations.

Results with J2 included are presented in Fig. 3(c). Asanticipated, J

xy

2 > 0 enhances the FM fluctuations in the ab

plane, which has two immediate effects: (i) The gab/gc ratiorequired to fit TX/TC decreases with increasing J2 and (ii)the behavior of χab for T > TC also approaches the one fromχ

expab as we increase J2, indicating that the effects of this extra

exchange term are considerably felt for T � TC. [For J2 >

0.5J1(1 + �) we have a Curie-Weiss temperature ab > 0.]Whereas for FM in-plane correlation χab(T ) shows a

distinct dependence on the anisotropy �1 (Fig. 2), this effectis considerably weaker for AFM correlations since in this caseχab is itself strongly suppressed. Therefore, we obtain similarresults, as in Fig. 3(c), for �1 = −0.10 and even �1 = −0.50.While this diminishes our predictive power with respect tothe precise value of the anisotropy, we recall that Fig. 1(b)suggests proximity to the bicritical point, which constrains �1

to be small.The good match of theoretical and experimental suscep-

tibilities in Fig. 3(c) suggests that our classical anisotropicHeisenberg model captures the essentials of the ferromagneticordering process in Yb(Rh0.73Co0.27)2Si2.

We note that the scenario of different competingkinds of in-plane exchange interactions is also (indirectly)supported by the complicated propagation vectors ob-served in pure YbCo2Si2, where �Q = 2π (1/4,0.08,1) and

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COMPETING ORDERS, COMPETING ANISOTROPIES, AND . . . PHYSICAL REVIEW B 90, 075138 (2014)

�Q = 2π (1/4,1/4,1) for the high-T and low-T phases, respec-tively. While such �Q might also result from competition ofpurely AFM exchanges, our analysis of the T dependence ofχab for T > TC gives a very strong hint that some of thesein-plane exchanges have to be ferromagnetic.

IV. MULTICRITICALITY: CLASSICAL AND QUANTUM

The modeling of Yb(Rh0.73Co0.27)2Si2 presented so farsuggests the relevance of bicritical behavior, arising fromthe competition of in-plane antiferromagnetism and out-of-plane ferromagnetism. Indeed, the temperature-doping phasediagram of YbRh2Si2 (Fig. 1) indicates the presence of a finite-temperature bicritical point at a doping level of x ≈ 25%. Inthe simplest scenario, the lower-temperature phase transitionline emerging from this point, labeled TL, would then cor-respond to a first-order transition between antiferromagneticand ferromagnetic phases, but more complicated structureswith mixed (e.g., canted) orders are possible as well (those arenot described by the simple Heisenberg models of Sec. II).

A remarkable feature of the TN and TL lines in Fig. 1 is thatTL, although more strongly suppressed with decreasing x ascompared to TN, appears to be finite almost down to x = 0.Given that TN is almost vanishing there as well, this invitesspeculations about (approximate) quantum bicriticality nearx = 0. This scenario would involve simultaneous quantumcriticality of both itinerant in-plane antiferromagnetism andc-axis ferromagnetism. In fact, some observations in stoichio-metric (or slightly Ge-doped) YbRh2Si2 have been suggestedto be compatible with ferromagnetic quantum criticality. Thisincludes the T −0.7 dependence of the Gruneisen parameter� and the T −0.3 dependence of the specific heat C/T at lowtemperatures above TN (see Ref. [2]). However, this agreementhas mainly been considered fortuitous, as the ordered phaseof stoichiometric YbRh2Si2 is not ferromagnetic. Quantum bi-criticality would thus provide a new angle on the observations.

On the theoretical side, we note that bicriticality involvingin-plane and c-axis order by itself is not exotic but simplycorresponds to a point with an emergent higher symmetry,here O(3). What is, however, interesting and potentially exoticis quantum bicriticality involving itinerant antiferromagnetismand ferromagnetism, which has not been studied theoretically.Within the Landau-Ginzburg-Wilson description, pioneeredby Hertz [17], these two transitions have different dynamicalexponents, z = 2 and 3, respectively, such that one canexpect rich and nontrivial crossover phenomena, even if bothtransitions are above their respective upper-critical dimension[18]. This defines a fascinating field for future theoreticalstudies [19].

We note that another possible candidate for such a quantumbicriticality scenario in an itinerant system is the Laves-phaseNbFe2, where competing AFM and FM orders have beenobserved near the quantum critical point [20].

V. COMPARISON TO YbNi4P2

Our success in modeling the finite-temperature suscepti-bilities of Yb(Rh0.73Co0.27)2Si2 suggests we should look atother materials with similar phenomenology. Remarkably, theheavy-fermion ferromagnet YbNi4P2 has been recently found

[10,11] to display a switch of magnetic response anisotropysimilar to Yb(Rh0.73Co0.27)2Si2.

YbNi4P2 is a tetragonal metal with a Curie temperatureof TC = 0.15 K. The low-temperature ferromagnetic momentpoints perpendicular to the c axis, as indicated by the divergentsusceptibility χab at TC. Notably, χc(T ) is larger than χab(T )for essentially all temperatures above TC. Hence, the g factorsobey gc > gab, such that both the exchange and g-factoranisotropies appear opposite to those of Yb(Rh0.73Co0.27)2Si2.In YbNi4P2, these anisotropies lead to a crossing of χab(T )and χc(T ) at TX/TC ≈ 1.13.

There is, however, an additional ingredient relevant forYbNi4P2 [in contrast to Yb(Rh0.73Co0.27)2Si2]. This materialappears to be located extremely close to a quantum criticalpoint. TC can be suppressed with a small amount of Asdoping in YbNi4(P1−xAsx)2, with TC → 0 extrapolated forx ≈ 0.1. This nearly quantum critical behavior in YbNi4P2 ismanifest in the temperature dependence of χ , which followsχc ∝ T −0.66 in the temperature range TC < T < 10 K. It is thispower law which cannot be reproduced by any means in ourclassical simulation. As seen in Fig. 2 the high-temperaturebehavior of χ is Curie-like, and this divergence becomesstronger upon approaching TC. In contrast, in YbNi4P2 thehigh-temperature Curie-like behavior, χc ∝ 1/T , is followedby the weaker power-law divergence upon cooling.

Recalling the good match we were able to find forYb(Rh0.73Co0.27)2Si2, we conclude that the latter materialis dominated by effectively classical magnetism, and thequantum critical behavior of stoichiometric YbRh2Si2 doesnot appear to extend to up to 27% of Co doping.

VI. SUMMARY

Motivated by the doping and temperature evolution of mag-netism of Yb(Rh1−xCox)2Si2, we studied Heisenberg modelswith competing anisotropies in the exchange interactions and g

factors. Such models possess ordered phases with spins alignedeither along the c axis or in the ab plane, separated from each atlow temperature by a first-order line which ends at a bicriticalpoint. The proximity to this bicritical point, combined withthe competition between ferromagnetic and antiferromagneticordering, naturally explains the “switch” in the anisotropy inthe magnetic susceptibilities, as experimentally observed inYb(Rh0.73Co0.27)2Si2.

While our classical local-moment modeling gives a goodaccount of the susceptibility data of Yb(Rh0.73Co0.27)2Si2near its finite-temperature phase transition, it fails to describeYbNi4P2, where a similar switch of response anisotropies hasbeen observed. This highlights the importance of quantumfluctuations, as YbNi4P2 is located very close to a putativeferromagnetic quantum critical point.

More broadly, our results point towards bicritical behaviorbeing relevant in certain Yb-based heavy-fermion compounds.Metallic quantum bicritical points, in particular involvingphenomena with different critical exponents, are thus anexciting field for future research.

ACKNOWLEDGMENTS

We are grateful to S. Lausberg for providing experi-mental data and for discussions. E.C.A., moreover, thanks

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ANDRADE, BRANDO, GEIBEL, AND VOJTA PHYSICAL REVIEW B 90, 075138 (2014)

0 1 2 3 4 50

0.51.01.52.02.53.0

T/J

(a) Paramagnetic

c axisab plane

−5 −4 −3 −2 −1 0D/J

0

0.5

1.0

1.5

2.0

T/J

(b) Paramagnetic

c axisab plane

FIG. 4. (Color online) Phase diagram of model (A1) for fixedvalues of the exchange anisotropy: � = 0.5 in (a) and � = −0.5in (b). The squares represent second-order phase transitions and thecircles first-order phase transitions between the two ordered states.The first order line ends at a bicritical point located at D = 2.220(2)(a) and D = −1.995(5) (b).

F. A. Garcia for discussions. This research was supported bythe DFG through FOR 960 and GRK 1621. E.C.A. was alsopartially supported by FAPESP. We also acknowledge ZIH atTU Dresden for the allocation of computing resources.

APPENDIX: MODEL WITH SINGLE-ION ANISOTROPY

To further explore the physics of bicriticality in a local-moment model, it is instructive to consider, in addition tothe exchange anisotropy, a single-ion anisotropy of the form−D

∑i(S

zi )2. While such a term is absent from a spin-1/2

model, it is generically present for spins S � 1 in a tetragonalenvironment.

For illustration, we consider a model with ferromagneticnearest-neighbor exchange only, with the Hamiltonian

H = −∑

〈i,j〉

∑

α

J αSαi Sα

j − D∑

i

(Sz

i

)2, (A1)

where D > 0 (D < 0) favors spin alignment along the c axis(in the ab plane). For the exchange anisotropy, we stick to theparametrization J x = J y = J (1 + �) and J z ≡ J . Studies ofthe model (A1) have appeared before in the literature [21,22],and we complement these results here.

1. Phase diagram

The competition between the single-ion anisotropy D andthe exchange anisotropy � in Eq. (A1) leads to phase diagramsas shown in Fig. 4. (Figure 4(b) matches within error barsthe corresponding result in Ref. [21].) At T = 0 there is afirst-order phase transition between in-plane and c-axis order atD = zJ�/2, where z is the lattice coordination number. Thistransition continues to finite T and terminates at at bicriticalpoint [22–24]. To efficiently sample all spin configurationsclose to this first-order phase transition, we employ theparallel-tempering algorithm [25,26]. The thermal transitionat the bicritical point is in the 3d Heisenberg universalityclass, i.e., the O(3) symmetry is restored here. We noticethat when crossing the first-order line upon heating up thesystem, we always favor the state which is the ground state forD = 0, which has a higher entropy because both spin anglesare unlocked.

2. Susceptibilities

We now turn to the behavior of the anisotropic magneticsusceptibilities (Fig. 5). Motivated by the experimental resultsto Yb(Rh0.73Co0.27)2Si2, we only consider magnetic order

10−1

100

101

102

χ( T

)

(a) χc

χab

(b) (c)

100 101

T/J

10−1

100

101

102

χ(T

)

(d)

100 101

T/J

(e)

100 101

T/J

(f)

FIG. 5. (Color online) Anisotropic susceptibilities χab and χc on a log-log scale for gab = gc and L = 32: (a) � = 0.5 and D = 2.20J ;(b) � = 0.5 and D = 2.25J ; (c) � = 0.5 and D = 2.50J ; (d) � = 1.0 and D = 4.40J ; (e) � = 1.5 and D = 6.60J ; and (f) � = −0.5 andD = −1.50J .

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along the c axis. For all results in Fig. 5, we consider gab = gc

and the effects of the single-ion anisotropy are encoded by theparameter D. In Figs. 5(a)–5(c) we see that as we approachthe bicritical point, the fluctuations in χab are enhanced.Interestingly, in 5(a) and 5(b), we even have χab > χc atintermediary temperatures above TC, with the two curvescrossing only very close to the transition. This differencebetween χab and χc is further enhanced if we increase thesingle-ion anisotropy D [Figs. 5(d) and 5(e)]. (In both of themwe keep the ratio D/� the same as in Fig. 5(a) so we do

not move too far away from the bicritical point.) Finally, inFig. 5(f) we consider both D and � negative and we alwayshave χc > χab for T > TC.

Therefore, we see that model (A1) naturally describes thecrossing of the susceptibilities above TC as a result of theproximity to a bicritical point. While it is true that the splittingbetween χab and χc above the crossing point is small, it is stillremarkable that a purely local-moment description is able tocapture such crossover, an observation which motivates us toassociate this crossing to the proximity of a bicritical point.

[1] P. Gegenwart, Q. Si, and F. Steglich, Nat. Phys. 4, 186 (2008).[2] H. von Lohneysen, A. Rosch, M. Vojta, and P. Wolfle, Rev. Mod.

Phys. 79, 1015 (2007).[3] O. Trovarelli, C. Geibel, S. Mederle, C. Langhammer, F. M.

Grosche, P. Gegenwart, M. Lang, G. Sparn, and F. Steglich,Phys. Rev. Lett. 85, 626 (2000).

[4] P. Gegenwart, J. Custers, C. Geibel, K. Neumaier, T. Tayama,K. Tenya, O. Trovarelli, and F. Steglich, Phys. Rev. Lett. 89,056402 (2002).

[5] S. Friedemann, T. Westerkamp, M. Brando, N. Oeschler,S. Wirth, P. Gegenwart, C. Krellner, C. Geibel, and F. Steglich,Nat. Phys. 5, 465 (2009).

[6] C. Klingner, C. Krellner, M. Brando, C. Geibel, F. Steglich,D. V. Vyalikh, K. Kummer, S. Danzenbacher, S. L. Molodtsov,C. Laubschat, T. Kinoshita, Y. Kato, and T. Muro, Phys. Rev. B83, 144405 (2011).

[7] P. Gegenwart, J. Custers, Y. Tokiwa, C. Geibel, and F. Steglich,Phys. Rev. Lett. 94, 076402 (2005).

[8] S. Lausberg, A. Hannaske, A. Steppke, L. Steinke, T. Gruner,L. Pedrero, C. Krellner, C. Klingner, M. Brando, C. Geibel, andF. Steglich, Phys. Rev. Lett. 110, 256402 (2013).

[9] C. Klingner, C. Krellner, M. Brando, C. Geibel, and F. Steglich,New J. Phys. 13, 083024 (2011).

[10] C. Krellner, S. Lausberg, A. Steppke, M. Brando, L. Pedrero,H. Pfau, S. Tence, H. Rosner, F. Steglich, and C. Geibel, New J.Phys. 13, 103014 (2011).

[11] A. Steppke, R. Kuchler, S. Lausberg, E. Lengyel, L. Steinke,R. Borth, T. Luhmann, C. Krellner, M. Nicklas, C. Geibel,F. Steglich, and M. Brando, Science 339, 933 (2013).

[12] J. Sichelschmidt, V. A. Ivanshin, J. Ferstl, C. Geibel, andF. Steglich, Phys. Rev. Lett. 91, 156401 (2003).

[13] D. P. Landau and K. Binder, A Guide to Monte Carlo Simulationsin Statistical Physics, 3rd ed. (Cambridge University Press,Cambridge, 2009).

[14] A nice example is the itinerant helimagnet MnSi, which canbe well described using local-moment models, as shown inS. Buhrandt and L. Fritz, Phys. Rev. B 88, 195137 (2013).

[15] T. Gruner, J. Sichelschmidt, C. Klingner, C. Krellner, C. Geibel,and F. Steglich, Phys. Rev. B 85, 035119 (2012).

[16] C. L. Henley, Phys. Rev. Lett. 62, 2056 (1989).[17] J. A. Hertz, Phys. Rev. B 14, 1165 (1976).[18] T. Meng, A. Rosch, and M. Garst, Phys. Rev. B 86, 125107

(2012).[19] In layered metals, quantum (bi)criticality may be accompanied

by dimensional crossover phenomena: M. Garst, L. Fritz,A. Rosch, and M. Vojta, Phys. Rev. B 78, 235118(2008).

[20] D. Moroni-Klementowicz, M. Brando, C. Albrecht, W. J.Duncan, F. M. Grosche, D. Gruner, and G. Kreiner, Phys. Rev.B 79, 224410 (2009).

[21] R. T. S. Freire, J. A. Plascak, and B. V. d. Costa, Braz. J. Phys.34, 452 (2004).

[22] W. Selke, Phys. Rev. E 87, 014101 (2013).[23] Y. Shapira and S. Foner, Phys. Rev. B 1, 3083 (1970).[24] W. Selke, Phys. Rev. E 83, 042102 (2011).[25] K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. 65, 1604

(1996).[26] C. E. Fiore, Phys. Rev. E 78, 041109 (2008).

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